A reproducing kernel Hilbert space method for nonlinear partial differential equations: applications to physical equations

Abstract

The partial differential equations (PDEs) describe several phenomena in wide fields of engineering and physics. The purpose of this paper is to employ the reproducing kernel Hilbert space method (RKHSM) in obtaining effective numerical solutions to nonlinear PDEs, which are arising in acoustic problems for a fluid flow. In this paper, the RKHSM is used to construct numerical solutions for PDEs which are found in physical problems such as sediment waves in plasma, sediment transport in rivers, shock waves, electric signals' transmission along a cable, acoustic problems for a fluid flow, vibrating membrane, and vibrating string. The RKHSM systematically produces analytic and approximate solutions in the form of series. The convergence analysis and error estimations are discussed to prove the applicability theoretically. Three applications are tested to show the performance and efficiency of the used method. Computational results indicated a good agreement between the exact and numerical solutions.

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

“…Holding an important position in various fields including but not limited to elasticity, heat transfer, dynamics, electromagnetic theory as well as quantum mechanics, PDEs have become the subject matter of current research. Accordingly, the subsequent paper is concerned with the PDEs describing several phenomena in broad range of areas like physics and engineering [11]. The authors of the work aim at employing the reproducing kernel Hilbert space method (RKHSM) for obtaining effective numerical solutions to nonlinear PDEs that arise in acoustic problems for a fluid flow.…”

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

“…In the last decade, the Reproducing Kernel Method (RKM) has been considered a solution for different differential equations, systems of differential equations, and partial differential equations, [18][19][20]. For example, the RKM has been used to solve nonlinear singular boundary value problems, nonlinear C-q-fractional Initial Value Problems (IVPs), linear systems of second-order boundary value problems, solve coupled systems of fractional order, systems of linear Volterra integral and equations, systems of linear Volterra integral equations with variable coefficients [21][22][23][24][25][26].…”

“…">IntroductionSome natural phenomena can be modeled by linear and non-linear systems of integral and differential equations, which are commonly used in fields such as biology, chemistry, and physics [1][2][3][4][5][6][7][8][9][10][11]. Many numerical methods, such as collocation boundary value methods, discontinuous Galerkin approximations, Euler matrix method, spectral element method, Chebyshev wavelets approach, and Radial basis Functions, have been provided to solve linear and nonlinear Volterra integral equations [12][13][14][15][16][17].In the last decade, the Reproducing Kernel Method (RKM) has been considered a solution for different differential equations, systems of differential equations, and partial differential equations, [18][19][20]. For example, the RKM has been used to solve nonlinear singular boundary value problems, nonlinear C-q-fractional Initial Value Problems (IVPs), linear systems of second-order boundary value problems, solve coupled systems of fractional order, systems of linear Volterra integral and equations, systems of linear Volterra integral equations with variable coefficients [21][22][23][24][25][26].The RKM offers several advantages, including its ease of implementation and ability to provide reasonably accurate approximations.…”

Uncertain M-fractional differential problems: existence, uniqueness, and approximations using Hilbert reproducing technique provisioner with the case application: series resistor-inductor circuit

“…The HRT, a powerful mathematical tool, has made significant discoveries in various stochastic physics and nonlinear frameworks [10][11][12]. Built upon the concept of Hilbert space, which provides a framework for analyzing functions and their properties, this algorithm has found applications in signal processing, probability, modeling, and control theory [13][14][15][16][17][18][19][20][21][22]. It has proven particularly useful in solving problems related to function approximation, interpolation, and regression.…”